{
 "metadata": {
  "name": "Power_Law_Distributions"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h1>Power Law Distributions</h1>\n",
      "We consider here probability distributions of the form \n",
      "\n",
      "$p(x) = \\frac{\\alpha}{x^{\\alpha+1}}$\n",
      "\n",
      "where $x \\ge x_{min} \\gt 0$ and $\\alpha \\gt 0$, known as a power law distribution. Such distributions are interesting because they break many significant assumptions used in standard classical statistics. For example, the first moment, or mean, of this distribution is undefined **when $\\alpha \\le 1$**:\n",
      "\n",
      "$E(x) = \\int_{x_{min}}^{\\infty} x \\frac{\\alpha}{x^{\\alpha+1}} = \\alpha \\frac{x^{1-\\alpha}}{1-\\alpha}\\vert_{x_{min}}^{\\infty}$\n",
      "\n",
      "for $\\alpha \\lt 1$ and\n",
      "\n",
      "$E(x) = \\alpha \\ln x \\vert_{x_{min}}^{\\infty}$\n",
      "\n",
      "for $\\alpha = 1$, both of which fail to converge in the limit as $x \\rightarrow \\infty$. Indeed, **all moments, $m \\ge \\alpha$, of this distribution fail to exist**. In such cases where the mean and standard deviation are not finite, the conditions of the *Central Limit Theorem* are not satisfied.\n",
      "\n",
      "<h2>Generating Power Law Distributed Data</h2>\n",
      "In principle, we can generate data following any given probability distribution, $p(x)$ using the following approach. \n",
      "\n",
      "1. Formulate the *cummulative distribution function* (CDF), $P(x) = y = \\int_{x_{min}}^x p(t)dt$. This is the probability that the random variable with distribution $p(x)$ will be less than or equal to $x$. By definition this function is monotonically increasing and takes values in the interval $[0,1]$.\n",
      "\n",
      "2. Formulate the *inverse* of the CDF, $x = P^{-1}(y)$. For a given value of $y \\in [0,1]$, the inverse CDF gives the value of $x$ where the CDF is equal to $y$. \n",
      "\n",
      "3. Generate sample values, $Y = \\\\{y_1, \\ldots, y_N\\\\}$ from a uniform distribution over $[0,1]$ and compute the inverse CDF for each. This will yield a set of values that are distributed according to the original distribution, $p(x)$\n",
      "\n",
      "For the power law distribution, the CDF is given by\n",
      "\n",
      "$P(x) = y = \\frac{1}{x_{min}^\\alpha} - \\frac{1}{x^\\alpha}$\n",
      "\n",
      "Letting $\\gamma = \\frac{1}{x_{min}^\\alpha}$, the inverse CDF is given by\n",
      "\n",
      "$x = \\frac{1}{(\\gamma - y)^{1/\\alpha}}$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h2>Example</h2>\n",
      "Generating artificial power law data"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "from matplotlib import pyplot as plt\n",
      "#CDF inverse\n",
      "#Recall, we REQUIRE alpha > 0\n",
      "def inv_cdf(y, xmin = 1.0, alpha = 1.0):\n",
      "    if not (y >=0 and y<1): raise Exception(\"y must be in the range [0,1]\")\n",
      "    if not alpha > 0: raise Exception(\"alpha parameter must be greater than zero\")\n",
      "    if xmin == 1.0: gamma = 1.0\n",
      "    else: gamma = 1.0 / math.pow(xmin,alpha)\n",
      "    return 1.0 / pow(gamma - y, 1.0/alpha)\n",
      "  \n",
      "def gen_data(N, xmin = 1, alpha = 1.0, invf = 10000):\n",
      "    means = []\n",
      "    maxs = []\n",
      "    s = 0\n",
      "    m = -1\n",
      "    idx = 0\n",
      "    maxy = 0\n",
      "    maxn = 0\n",
      "    while idx < N:\n",
      "        y = np.random.uniform()\n",
      "        if y > maxy: \n",
      "            maxy = y\n",
      "            maxn = idx\n",
      "        x = inv_cdf(y, xmin, alpha)\n",
      "        s += x\n",
      "        m = max(m,x) #current max in data\n",
      "        if idx%invf == 0:\n",
      "            means.append(s/(idx+1.0))\n",
      "            maxs.append(m)\n",
      "        idx += 1\n",
      "    return (means, maxs, maxy, maxn)\n",
      "\n",
      "def _plot(ax, data, xlab, ylab, title, log=True):\n",
      "    if log: ax.semilogy(range(len(data)),data)\n",
      "    else: ax.plot(range(len(data)),data)\n",
      "    ax.set_ylabel(ylab)\n",
      "    ax.set_xlabel(xlab)\n",
      "    ax.set_title(title)\n",
      "    ax.xaxis.grid(color='gray', linestyle='dashed')\n",
      "    ax.yaxis.grid(color='gray', linestyle='dashed')\n",
      "    ax.set_xticklabels([])\n",
      "\n",
      "f, axarr = plt.subplots(2,2)\n",
      "f.subplots_adjust(right=1.5)\n",
      "f.subplots_adjust(top=2.5)\n",
      "N = 1e6\n",
      "alpha = 1.0\n",
      "means1, maxs1, my, mn = gen_data(N, 1.0, alpha)\n",
      "_plot(axarr[0][0], means1, '', 'Mean', 'alpha = {0}'.format(alpha), False)\n",
      "_plot(axarr[0][1], maxs1, '', 'Max', 'alpha = {0}'.format(alpha))\n",
      "print 'For alpha = {0}, the max y = {1} at n = {2}'.format(alpha,my,mn)\n",
      "alpha = 1.4\n",
      "means2, maxs2, my, mn = gen_data(N, 1.0, alpha)\n",
      "print 'For alpha = {0}, the max y = {1} at n = {2}'.format(alpha,my,mn)\n",
      "_plot(axarr[1][0], means2, '', 'Mean', 'alpha = {0}'.format(alpha), False)\n",
      "_plot(axarr[1][1], maxs2, '', 'Max', 'alpha = {0}'.format(alpha))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "For alpha = 1.0, the max y = 0.999995453705 at n = 897085\n",
        "For alpha = 1.4, the max y = 0.999999280851 at n = 7088"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n"
       ]
      },
      {
       "output_type": "display_data",
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SxGKPiIjayzVpAjz5JPCvfwFDh9avp2/vXmDKFDGc7Ktp08Rwc06OWJCSkiLO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\nDJDuvVf65S+lujrpmWfK+/MB4P1SqZQGDkyaWtIFED1vj1bLJ65nVg4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+AAAB\na0lEQVRqamoq148DgLLpaHxrb29XQ0ODmpqatHr1ajU2NmrNmjWqqqpSa2urJGnHjh2dfubgwSVt\nMoBupqxHq2UyGU2dOlVPPfXU3g1JJMrVDACG+HK02vvHtz/84Q+aN2/ezkLwmmuukSR96UtfUkND\ng3r37q3Jkyervr5+r89iPARQjK7Gw4oytqNLvgzaABCF3ZduJamqqkorVqxQ37599dOf/rTL72U8\nBBA1XtoAgBJglg6ATyj4AKAEKisrdz6rJ0mtra2qqqqKsUUAujMKPgAogQkTJmjdunXKZDJqa2vT\n4sWLNW3atLibBaCbKlvBV19fr0mTJmnt2rWqrq7WwoULy/WjAaCkOhrfKioqtGDBAk2ZMkU1NTWa\nPn26Ro0aFXdTAXRTZX1LFwAAAOXHki4AAIBxFHwAAADGUfABAAAYR8EHAABgHAUfAACAcRR8AAAA\nxlHwAQAAGEfBBwAAYBwFHwAAgHH/H/LpufPwjvrtAAAAAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    }
   ],
   "metadata": {}
  }
 ]
}